Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

A three‐dimensional hybrid meshfree‐Cartesian scheme for fluid–body interaction

Abstract: A numerical method based on a hybrid meshfree-Cartesian grid is developed for solving three-dimensional fluid–solid interaction (FSI) problems involving solid bodies undergoing large motion. The body is discretized and enveloped by a cloud of meshfree nodes. The motion of the body is tracked by convecting the meshfree nodes against a background of Cartesian grid points. Spatial discretization of second-order accuracy is accomplished by the combination of a generalized finite difference (GFD) method and conventional finite difference (FD) method, which are applied to the meshfree and Cartesian nodes, respectively. Error minimization in GFD is carried out by singular value decomposition. The discretized equations are integrated in time via a second-order fractional step projection method. A time-implicit iterative procedure is employed to compute the new/evolving position of the immersed bodies together with the dynamically coupled solution of the flow field. The present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow. Good agreement with published results shows the ability of the present hybrid meshfree-Cartesian grid scheme to achieve good accuracy. An application of the method to the self-induced propulsion of a deforming fish-like swimmer further demonstrates the capability and potential of the present approach for solving complex FSI problems in 3D. Copyright © 2011 John Wiley & Sons, Ltd.

Large deflection analysis of flexible plates by the meshless finite point method

Abstract: The classical finite difference technique and methods based on series expansions can only be adopted for solving plates with simple geometry, loading and boundary conditions. In contrast, the finite element method has been widely used for general analysis of bending and flexible plates (coupled bending and in-plane effects). Lack of stress continuity and relatively expensive mesh generation and remeshing schemes have led to the emergence of meshless methods, such as the finite point method (FPM). FPM is a strong form solution which combines the moving least square interpolation technique on a domain of irregularly distributed points with a point collocation scheme to derive system governing equations. In this study, coupled nonlinear partial differential equations of fourth order are solved to analyse large deflection behaviour of plates subjected to lateral and in-plane loadings. Several plate problems are solved and compared with analytical solution and other available numerical results to assess the performance of the proposed approach.

Numerical modeling of NPZ and SIR models with and without diffusion

Abstract: In this paper, the two biological models i.e. Nitrogen, Phytoplankton and Zooplankton (NPZ) and whooping cough SIR models (Charpentier et al., 2010) are being modified and solved numerically by finite difference and meshless methods. Diffusion process has been added to the existing models (Charpentier et al., 2010) so that a unidimensional movement of three species can be incorporated in the models. The effects of diffusion has been studied in both the models. An operator splitting method coupled with the meshless and finite difference procedures, is being considered for numerical solution of the two biological models with and without diffusion. A one step explicit meshless procedure is also applied for the numerical solution of the nonlinear models. The NPZ model contains the concentration of Nitrogen, Phytoplankton and Zooplankton and the whooping cough model contains susceptible, infected, and recovered classes of the population. Equilibrium points of both models have been investigated. Stability of equilibrium points regarding SIR model has been studied. The basic reproduction number of SIR model is also determined. Due to non-availability of the exact solution, the numerical results obtained are mutually compared and their correctness is being verified by the theoretical results as well.